https://file.aiquickdraw.com/mathbot/file/1727201671098ni4prfaw.jpg

The image you uploaded shows a graph of a function along with some questions about its key features such as relative maximum, minimum, intervals of increase and decrease, domain, and range. Based on the graph and the partially filled responses, here's what I can infer:

- **Relative Maximum**: The function reaches a relative maximum value of **7** at **x = 8**. This is correct as indicated in the image.
- **Relative Minimum**: The function has a relative minimum of **-1** at **x = 0**, which is also correct.

Now let's complete the rest:

### Intervals of Increase and Decrease:
1. The function **increases** on intervals where the slope of the graph is positive. From the graph, it increases from \( x = 0 \) to \( x = 8 \) because the curve is going upwards after reaching the minimum.
    - **Increasing interval**: \( (0, 8) \)

2. The function **decreases** on intervals where the slope is negative. It decreases from \( x = -\infty \) to \( x = 0 \) as the curve moves downwards before reaching the minimum.
    - **Decreasing interval**: \( (-\infty, 0) \) and after the maximum, it also decreases from \( x = 8 \) to \( \infty \).
    - So, **Decreasing interval** is: \( (-\infty, 0) \cup (8, \infty) \)

### The Domain and Range:
- The **domain** of the function is all real numbers, already filled as \( (-\infty, \infty) \), which is correct.
- The **range** of the function is from its minimum value (which is -1) to infinity because the graph continues upwards after reaching the relative maximum at 7.
    - **Range**: \( [-1, \infty) \)

Would you like further details on any part of the explanation?

Here are 5 questions to deepen understanding:
1. What defines a relative maximum or minimum on a graph?
2. How do you determine intervals of increase or decrease by looking at a graph?
3. Why is the domain of this function \( (-\infty, \infty) \)?
4. What is the significance of the range in relation to the graph's behavior?
5. How can you identify concavity from a graph, and why is that important?

**Tip**: For any continuous function, always check where the function turns from increasing to decreasing or vice versa to locate relative extrema (maxima or minima).

Consider the function in the graph to the right. The function has a relative maximum of 7 at x = 8 and a relative minimum of -1 at x = 0. Identify the intervals where the function is increasing and decreasing, along with the domain and range.

Learn how to analyze the relative maximum and minimum values of a function using a graphical representation. Identify the intervals where the function increases or decreases and determine the domain and range. This concept applies to students studying functions and calculus in Grades 9-12.

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